\(P_{\mathcal {W}(X)} T|_{\mathcal {W}(X)}\)  is a pure contraction on \(\mathcal {W}(X)\) .
Note that \((i)\)  implies \((ii)\)  follows in a manner that is similar to the above results.
\((ii) \Rightarrow (i):\)  From equation (REF ), it follows that
\(UTU^* (M_{z_i} \otimes I_{\mathcal {E}}) = (M_{z_i} \otimes I_{\mathcal {E}}) UTU^* \quad (i \in \lbrace 1,\ldots ,n\rbrace ).\) 
This implies that there exists a contractive multiplier \(\Theta \in \mbox{Mult}_1(H_m(\mathbb {B}_n, \mathcal {W}(X)))\)  such that \(T = U^* M_{\Theta } U\)  (see Proposition 4.2, [1]}). Now assumption \((b)\)  implies that \(\mathcal {W}(X) \subseteq \ker A_{T}\) , and therefore, for any \(\eta \in \mathcal {W}(X)\)  we have
\(\lim _{n \rightarrow \infty } \Vert M_{\Theta }^{*n} U \eta \Vert  = \lim _{n \rightarrow \infty } \Vert U T^{*n} \eta \Vert  = \lim _{n \rightarrow \infty } \Vert T^{*n} \eta \Vert  =0.\) 
Thus, \(\mathcal {W}(X) = U \mathcal {W}(X) \subseteq \ker A_{M_{\Theta }}\)  and therefore, Theorem REF  implies that \(M_{\Theta }\)  is a pure contraction on \(H_m(\mathbb {B}_n, \mathcal {W}(X))\) . And by unitary equivalence, we get that \(T\)  is also a pure contraction on \(\mathcal {H}\) . This completes the proof.

Applications
Recently, the author proved that pair of commuting pure contractions with finite dimensional defect spaces admit pure isometric dilation (see [2]}). The aim of this section is to apply Theorem REF  to extend the above result for certain important tuples of commuting pure contractions appearing in recent papers [3]} and [4]}.
Let
\(\mathcal {X}^n(\mathcal {H}) := \lbrace (X_1,\ldots ,X_n) \in \mathcal {B}(\mathcal {H}): \Vert X_ih\Vert  \le 1, X_iX_j = X_jX_i, 1 \le i,j \le n\rbrace \) 
A tuple \(X \in \mathcal {X}^n(\mathcal {H})\)  is said to be pure if \(X_i\)  is a pure contraction for all \(i \in \lbrace 1,\ldots ,n\rbrace \) .
With respect to the Szegö kernel \(\mathbb {S}_n(\mathbf {z}, \mathbf {w})\)  of the Hardy space \(H^2(n)\) , let
\(\mathbb {S}_n(\mathcal {H}): = \lbrace X \in \mathcal {X}^n(\mathcal {H}): \mathbb {S}_n^{-1}(X,X^*) \ge 0\rbrace .\) 
For the sequel, we need a special type of operator-valued multipliers called the Schur-Agler class on \(n\)  (see [5]}).
Definition 5.1 For a Hilbert space \(\mathcal {E}\) , the Schur-Agler class on \(n\)  (denoted by \(\mathcal {S}\mathcal {A}_n(\mathcal {E},\mathcal {E})\) ) is defined to be the collection of all \(\mathcal {B}(\mathcal {E})\) -valued bounded analytic function on \(n\)  for which there exists Hilbert spaces \(\lbrace \mathcal {H}_i\rbrace _{i=1}^n\)  and a unitary
\(U= \begin{bmatrix}A & B \\C & D\end{bmatrix}: \mathcal {E}\oplus (\underset{i=1}{\overset{n}{\oplus }} \mathcal {H}_i) \rightarrow \mathcal {E}\oplus (\underset{i=1}{\overset{n}{\oplus }} \mathcal {H}_i),\) 
such that
\(\Phi (\mathbf {z}) := A + B E(\mathbf {z})(I - DE(\mathbf {z}))^{-1}C \quad (\mathbf {z} \in n),\) 
where \(E(\mathbf {z}) := \oplus _{i=1}^n z_i I_{\mathcal {H}_i}\) .

It follows from a well-known computation that any element in \(\mathcal {S}\mathcal {A}_n(\mathcal {E},\mathcal {E})\)  is contractive.
Let us begin by observing a result that follows from [Theorem 4.2 and Proposition 4.1, [3]}]. For the sake of completeness we have included the proof of the result.
Lemma 5.2 ([3]},[4]}) 
Let \(T\)  be a contraction on a Hilbert space \(\mathcal {H}\)  for which there exists a Hilbert space \(\mathcal {E}\) , \(\Psi \in \mathcal {S}\mathcal {A}_n(\mathcal {E},\mathcal {E})\)  and an isometry \(\Pi :\mathcal {H}\rightarrow H_{\mathcal {E}}^{2}(n)\)  (where, \(n \ge 1\) ), such that \(\Pi T^* = M_{\Psi }^* \Pi \)  and \(H_{\mathcal {E}}^{2}(n) = \bigvee _{\mathbf {k} \in \mathbb {N}^n} M_{\mathbf {z}}^{\mathbf {k}} \Pi \mathcal {H}\) . If \(T\)  is a pure contraction on \(\mathcal {H}\) , then \(\Psi (0)\)  is a c.n.u. contraction on \(\mathcal {K}\) .

By definition, \(\Psi \in \mathcal {S}\mathcal {A}_n(\mathcal {E},\mathcal {E})\)  implies that there exist Hilbert spaces \(\lbrace  \mathcal {H}_i \rbrace _{i=1}^n\)  and a unitary \(U\)  of the form in equation (REF ) such that
\(\Phi (\mathbf {z}) = A + B E(\mathbf {z})(I - DE(\mathbf {z}))^{-1}C \quad (\mathbf {z} \in n).\) 
By the Nagy-Foias decomposition of a contraction there exists an orthogonal decompostion \(\mathcal {E}= \mathcal {E}_{0} \oplus \mathcal {E}_{1}\)  such that
\(A = \begin{bmatrix}A|_{\mathcal {E}_0} & 0 \\0 & A|_{\mathcal {E}_1}\end{bmatrix}: \mathcal {E}_0 \oplus \mathcal {E}_1 \rightarrow \mathcal {E}_0 \oplus \mathcal {E}_1,\) 
where, \(A|_{\mathcal {E}_{0}}\)  is a unitary and \(A|_{\mathcal {E}_1}\)  is a c.n.u. contraction. With respect to this decomposition of \(A\) , the transfer function \(\Psi (\mathbf {z})\)  can be decomposed in the following manner (see [Proposition 3.1, [4]}])
\(\Psi (z) = \begin{bmatrix}\Psi _{0}(\mathbf {z}) & 0 \\0 & \Psi _{1}(\mathbf {z})\end{bmatrix} \quad (\mathbf {z} \in n),\) 
where, \(\Psi _0(\mathbf {z}) = A|_{\mathcal {E}_0}\)  and \(\Psi _1(\mathbf {z}) = A|_{\mathcal {E}_1} + B E(\mathbf {z})(I - DE(\mathbf {z}))^{-1}C \in \mathcal {S}_n(\mathcal {E}_1,\mathcal {E}_1)\) . The proof will be complete if we can show that \(\Pi \mathcal {H}\subseteq H_{\mathcal {E}_{1}}^{2}(n)\) , since then \(H_{\mathcal {E}}^2({n}) = \bigvee _{\mathbf {k} \in \mathbb {N}^n} M_{\mathbf {z}}^{\mathbf {k}} \Pi \mathcal {H}\subseteq H_{\mathcal {E}_{1}}^{2}(n) \subseteq H_{\mathcal {E}}^2({n})\)  will imply that \(\mathcal {E}_0 = \lbrace 0\rbrace \) . Now \(g \in H_{\mathcal {E}_0}^2(n)\)  implies that there exist \(h_m \in H_{\mathcal {E}_0}^2(n)\)  for all \(m \in \mathbb {N}\)  such that \(g = M_{\Psi }^m h_m\)  (since \(M_{\Psi _0}\)  is a unitary on \(H_{\mathcal {E}_0}^2(n)\) ). Thus, for \(f \in \Pi \mathcal {H}\)  we have
\(| \langle f,g \rangle | = | \langle f, M_{\Psi }^{m} h_m \rangle | = | \langle M_{\Psi }^{*m} \Pi h, h_m \rangle | = | \langle \Pi X^{*m}h, h_m \rangle | \le \Vert X^{*n}h\Vert  \Vert g\Vert ,\) 
Taking limit on both sides as \(m \rightarrow \infty \)  gives \(\langle f,g \rangle = 0\)  which implies that \(\Pi \mathcal {H}\perp H_{\mathcal {E}_0}^2({n})\) . This completes the proof.

Tuples associated to von Neumann inequality in polydisc
Let us begin by giving a brief introduction on the tuples of operators appearing in [3]}. A particular class of these tuples were first studied by Grinshpan et al. ([11]}) in connection to multivariate von Neumann inequality. For each \(i, p, q \in \lbrace 1,\ldots ,n\rbrace \)  such that \(p \ne q\) , let
\(\hat{X}_i := \lbrace X \in \mathcal {X}^n(\mathcal {H}): (X_1,\ldots ,X_{i-1},X_{i+1},\ldots ,X_n) \in \mathcal {X}^{n-1}(\mathcal {H})\rbrace .\) 
and
\(\mathcal {X}_{p,q}^n(\mathcal {H}) := \lbrace X \in \mathcal {X}^n(\mathcal {H}): \hat{X}_p, \hat{X}_q \in \mathbb {S}_{n-1}(\mathcal {H}) \text{ and } \hat{X}_p \text{ is pure}\rbrace .\) 
\(X \in \mathcal {X}_{p,q}(\mathcal {H})\)  is said to be of finite rank if
\(\mathcal {D}_{\hat{X}_{i}} := \overline{\mbox{ran}} \hspace{2.84526pt}\mathbb {S}^{-1}(\hat{X}_i,{\hat{X}_{i}}^*)^{\frac{1}{2}},\) 
is a finite dimensional subspace of \(\mathcal {H}\) , for \(i=p,q\) .
The following result on isometric dilation for these tuples was obtained in [Theorem 5.3, [3]}].
Theorem 5.3 
Let \(X=(X_1,\ldots ,X_n) \in \mathcal {X}_{p,q}^n(\mathcal {H})\)  be finite rank, then there exists a Hilbert space \(\mathcal {E}\) , inner multiplies \(\Phi _p,\Phi _q \in H_{\mathcal {E}}^{\infty }({n-1})\) 
and an isometry \(\Pi : \mathcal {H}\rightarrow H_{\mathcal {E}}^2({n-1})\)  such that
\(\Pi X_i^* = \left\lbrace \begin{array}{ll}M_{z_i}^* \Pi \quad (i \in \lbrace 1,\ldots ,n-1\rbrace  \setminus \lbrace p,q\rbrace ),\\M_{\Phi _i}^* \Pi \quad (i=p,q),\end{array}\right.\) 
where \(\Phi _p,\Phi _q\)  are inner polynomials of degree atmost one such that
\(\Phi _p(\mathbf {z}) \Phi _q(\mathbf {z}) = \Phi _q(\mathbf {z}) \Phi _p(\mathbf {z}) = z_p I_{\mathcal {E}} \quad (\mathbf {z} \in {n-1}).\) 
In other words, \(X\)  dilates to a \(n\) -tuple of commuting isometries
\((M_{z_1},\ldots ,M_{z_{p-1}}, M_{\Phi _p},M_{z_{p+1},} \ldots , M_{z_{q-1}}, M_{\Phi _q}, M_{z_{q+1}}, \ldots ,M_{z_n})\) 
on \(H_{\mathcal {E}}^2({n-1})\) .

A remarkable contribution of the above result lies in the fact that \(M_{\Phi _p},M_{\Psi _q}\)  are explicitly determined to be of Berger, Coburn and Lebow type isometries [13]}. In particular, there exists a collection \((\mathcal {E},U,P)\)  where, \(\mathcal {E}\)  is a Hilbert space, \(U\)  is a unitary and \(P\)  is an orthogonal projection such that
\(\Phi _p(\mathbf {z}) = (P+z_pP^{\perp })U^*; \quad \Phi _q(\mathbf {z}) = U(P^{\perp } + z_p P).\) 
Using Theorem REF , we can establish the following result on pure isometric dilation.
Theorem 5.4 Let \(X=(X_1,\ldots ,X_n)\)  be a tuple of commuting pure contractions which belongs to the collection \(\mathcal {X}_{p,q}^n(\mathcal {H})\)  and has finite rank. Then there exists a Hilbert space \(\mathcal {E}\)  such that \(X\)  dilates to a \(n\) -tuple of commuting pure isometries
\((M_{z_1},\ldots ,M_{z_{p-1}}, M_{\Phi _p},M_{z_{p+1},} \ldots , M_{z_{q-1}}, M_{\Phi _q}, M_{z_{q+1}}, \ldots ,M_{z_n})\) 
on \(H_{\mathcal {E}}^2({n-1})\) .

Using Lemma REF  and Theorem REF  we obtain that \(\Phi _p(0), \Psi _{q}(0)\)  are c.n.u. contractions on the finite dimensional space \(\mathcal {E}\) , which further implies that \(\Phi (0),\Psi (0)\)  are pure contractions on \(\mathcal {E}\)  (see [14]}). Now, Theorem REF  implies that \(M_{\Phi _p}, M_{\Psi _q}\)  are pure isometries. This completes the proof.

Tuples associated to commutant lifting in unit polydisc
In the same manner as above, let us begin with a brief overview of tuples of contractions appearing in [4]}. These tuples were first studied by Ball et al. in their influential paper [16]} on the commutant lifting problem in the polydisc. Given \(A \in \mathcal {B}(\mathcal {H})\) , a conjugate map \(C_A:\mathcal {B}(\mathcal {H}) \rightarrow \mathcal {B}(\mathcal {H})\)  is the completely positive map defined by
\(C_A(Y) := AYA^* \quad (Y \in \mathcal {B}(\mathcal {H})).\) 
Definition 5.5 \(\mathcal {P}^n(\mathcal {H})\)  is defined to be the collection of \(n\) -tuples of operators \(X \in \mathcal {X}^n(\mathcal {H})\)  that satisfy the following conditions:

\(\hat{X}_n \in \mathbb {S}_{n-1}(\mathcal {H})\) ;

There exist positive operators \(G_1,\ldots ,G_{n-1} \in \mathcal {B}(\mathcal {H})\)  (depending on \(X\) ) for which

\(I - X_n X_n^* = G_1 + \ldots + G_{n-1}\) ;

\( S_X(G_i):= \prod _{j=1; j \ne i}^{n-1} (I_{\mathcal {B}(\mathcal {H})} - C_{X_j}) G_i \ge 0\)  for all \(i \in \lbrace 1,\ldots ,n\rbrace \) .

are satisfied.


The following result on isometric dilation of these tuples was obtained by Barik et al. in [Theorem 4.4, [4]}].
Theorem 5.6 
If \(X \in \mathcal {P}_n(\mathcal {H})\)  is of finite rank, then there exists an inner multiplier \(\Phi \in \mathcal {S}\mathcal {A}_{n-1}(\mathcal {D}_{\hat{X}_n},\mathcal {D}_{\hat{X}_n})\)  and an isometry \(\Pi : \mathcal {H}\rightarrow H_{\mathcal {D}_{\hat{X}_n}}^2({n-1})\)  such that
\(\Pi X_i^* = \left\lbrace \begin{array}{ll}M_{z_i}^* \Pi \quad (i=1,\ldots ,n-1),\\M_{\Phi }^* \Pi \quad (i=n).\end{array}\right.\) 
In other words, every finite rank \(X \in \mathcal {P}_n(\mathcal {H})\)  dilates to \((M_{z_1},\ldots ,M_{z_{n-1}}, M_{\Phi })\)  on \(H_{\mathcal {D}_{\hat{X}_n}}^2({n-1})\) .

The precise structure of the multiplier \(\Phi \in \mathcal {S}\mathcal {A}_{n-1}(\mathcal {D}_{\hat{X}_n}, \mathcal {D}_{\hat{X}_n})\)  is the following:
let \(F_i^2 := S_{X}(G_i)\)  for \(i=1,\ldots ,n-1\)  and \(\mathcal {F}_i = \overline{ran} F_i\) . The unitary in our discussion is \(U:\mathcal {D}_{\hat{X}_n} \oplus (\oplus _{i=1}^{n-1} \mathcal {F}_i) \rightarrow \mathcal {D}_{\hat{X}_n} \oplus (\oplus _{i=1}^{n-1} \mathcal {F}_i)\)  such that
\(U(D_{\hat{X}_n}h, F_1 X_1^*h,\ldots ,F_{n-1}X_{n-1}^* h) = (D_{\hat{X}_n}X_n^*h, F_1 h, \ldots , F_n h).\) 
Let
\(U= \begin{bmatrix}A & B \\C & D\end{bmatrix}: \mathcal {D}_{\hat{X}_n} \oplus (\oplus _{i=1}^{n-1} \mathcal {F}_i) \rightarrow \mathcal {D}_{\hat{X}_n} \oplus (\oplus _{i=1}^{n-1} \mathcal {F}_i),\) 
and \(E(\mathbf {z}) := \oplus _{i=1}^{n-1} z_i I_{\mathcal {F}_i} \in \mathcal {B}(\oplus _{i=1}^{n-1} \mathcal {F}_{i})\) . Then \(\Phi (\mathbf {z})\)  has the realization in the following manner,
\(\Phi (\mathbf {z}) := \tau _{U}(\mathbf {z}) = A + B E(\mathbf {z})(I - DE(\mathbf {z}))^{-1}C \quad (\mathbf {z} \in n),\) 
Using Theorem REF , we obtain the following result on pure isometric dilation.
Theorem 5.7 Let \(X=(X_1,\ldots ,X_n)\)  be a tuple of commuting pure contractions which belongs to the collection \(\mathcal {P}_n(\mathcal {H})\)  and has finite rank. Then there exists a pure inner multiplier \(\Psi (z) \in \mathcal {S}\mathcal {A}_{n-1}(\mathcal {D}_{\hat{X}_n},\mathcal {D}_{\hat{X}_n})\)  such that \(X\)  dilates to a \(n\) -tuple of commuting pure isometries \((M_{z_1}, \ldots , M_{z_{n-1}}, M_{\Phi })\)  on \(H_{\mathcal {D}_{\hat{X}_{n}}}^2({n-1})\) .

Using Lemma REF  and Theorem REF  we obtain that \(\Phi (0)\)  is a c.n.u. contraction on the finite dimensional Hilbert space \(\mathcal {D}_{\hat{X}_{n}}\)  and therefore, \(\Phi (0)\)  is a pure contraction on \(\mathcal {D}_{\hat{X}_{n}}\) . Now, Theorem REF  shows that \(M_{\Phi }\)  is a pure isometry on \(H_{\mathcal {D}_{\hat{X}_n}}^2({n-1})\) . This finishes the proof.

Acknowledgements
The author would like to thank Prof. B.K. Das for useful discussions and he is grateful to Prof. Kunyu Guo for a correspondence regarding Proposition REF . He would also like to thank Prof. Gautam Bharali for going through the initial drafts of this article and providing important feedback. The author is supported by DST-INSPIRE Faculty Fellowship No. - DST/INSPIRE/04/2019/000769.
